(This talk was given as part of the What We Talk About lecture series at No One Writes to the Colonel on May 17, 2012)
Math is many things: beautiful, fundamental, universal, but ultimately, math is hard. So hard in fact that mathematicians often need to phone up their other mathematician friends for help. The idea of the crazy-maned recluse furiously working in his office alone is an outdated one. While some modern mathematicians still fit this bill, collaboration is increasingly the norm. By the year 2000, the number of mathematics papers with a single author had shrunk to 50%. More and more people are tackling difficult math problems as a team.
No one embodies the idea of mathematical collaboration more than Paul Erdös (pronounced err-desh or air-dish), a Hungarian mathematician who lived in the twentieth century. A legendary figure in mathematics, Erdös published around 1500 papers and had around 500 co-authors. To contrast, most mathematicians write 7 papers in their entire life! Erdös was heavily in support of working together to solve math problems and questions, and also had incredible mathematical taste. He asked very interesting questions and would often attach a dollar amount to the questions. If you were clever enough to solve one of these Erdös questions, Paul Erdös himself would send you a cheque. These cheques were so revered in mathematics that often people frame them rather than cash them. More information on his very interesting life is available here.
It is in this context that the idea of the Erdös number emerged. In the 60s, one of Erdös’ friends presented the idea of the Erdös number to measure how close you were to Erdös in terms of collaboration. So Erdös himself has Erdös number 0 and all of his 511 direct co-authors have Erdös number 1. Now if you co-author a paper with say, Terrence Tao who has Erdös number 1, you would have Erdös number 2! If, like me, you have never published a paper, or you have never co-authored a paper, you would have an infinite Erdös number.
I would say that most mathematicians know their Erdös number (or can easily find it out); it is kind of like a feather in their hat. Your Erdös number on its own won’t get you a job, a diploma or a date, but it does look nice. Some people have been known to write papers with collaborators simply to lower their Erdös number (I think someone was selling the opportunity to co-author with them on Ebay) but this is relatively rare.
Now for some facts (all of which I found on the helpful Erdös Number Project). The smallest Erdös number is of course 0, but the largest is 13 (or 15 depending on what you consider to be a paper). Arguably the most famous mathematician, Andrew Wiles who solved Fermat’s Last Theorem, has Erdös number 3. Looking at the who’s who of brilliant mathematicians we see that every Fields medalist (kind of the math Nobel prize) has had Erdös number 5 or less, with most having number 2 or 3. In contrast the upper bound for the Nobel Prize in Medicine is 11.
Many non-mathematicians have Erdös numbers as well. Bill Gates, Stephen Hawking and Carl Sagan each have number 4. Some fields are even known for having low Erdös numbers. Biology and Linguistics are examples, as in Linguistics, Noam Chomsky has Erdös number 4.
Even more surprisingly, there are many actors with Erdös numbers. Natalie Portman (5) and Colin Firth (6) are counted among the various co-co-co-co-collaborators of Erdös. This leads to the inevitable connection with the “6 degrees of Kevin Bacon” game. You remember the game where I give you an actor and you try to connect them to Kevin Bacon by saying “A was in a film with B who was in a film with C … who was in a film with Kevin Bacon”.
Thus you have an actor’s “Bacon number”, the length of the shortest path of that sort to Kevin Bacon. Going even further you have “Erdös-Bacon numbers” (it’s a real thing, look it up). This is the sum of a person’s Erdös number and their Bacon number. Of the people I mentioned, Carl Sagan has EB number 7, Stephen Hawking has EB number 7 (if you count his appearance on the Simpsons), Colin Firth has EB number 7 and Natalie Portman has EB number 6. The smallest (non-disputed) EB number is 3 which belongs to Bruce Reznick a math professor with Erdös number 1 that appeared as an extra in Pretty Maids all in a Row, which gives him Bacon number 2.
By the way, this is from the Wikipedia plot synopsis of Pretty Maids All in a Row: “The story is set in Oceanfront High School, a fictitious American high school in the height of the sexual revolution. Young female students are being targeted by an unknown serial killer. Meanwhile, a male student called Ponce is experiencing sexual frustration, surrounded by a seemingly unending stream of beautiful and sexually provocative classmates.”
To end, let’s examine the famous baseball player Hank Aaron (who in my mind is still the all time leader in home runs) who has Bacon number 2 (he was in “Summer Catch” with Susan Gardner who was in “In the Cut” with Kevin Bacon). Hank Aaron has also signed the same baseball as Paul Erdös, giving him a tenuous Erdös number 1. So you could say that Hank Aaron also has an Erdös-Bacon number of 3!
Mike Pawliuk - Mathematics 
I wonder about the statement “most mathematicians write 7 papers in their entire life!” More likely you mean that the average lifetime number of papers per mathematician is (about) 7. I find it hard to believe that more mathematicians write exactly 7 papers than all other numbers of papers combined. Sorry to nitpick but it seems hard to let this pass!
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Yes, you’re right. Most mathematicians who write a paper write exactly 1 or 2 papers. Here is the fact that I loosely adapted.
From The Erdos number project:
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Nice post, Mike.
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Well written, Michael, especially:
“Many non-mathematicians have Erdös numbers as well.”
and…
“Hank Aaron has also signed the same baseball as Paul Erdös, giving him … Erdös number 1.”
This may make mine arguably 2, from Conway (J. Horton) and Graham (Ron), or 4 from Nash (J. Forbes), although more likely infinite, if not non-existant-zero, not to be confused with Paul’s existant-zero.
An as yet unpublished article Conway asked me to write, later led me to correspond briefly with Professor Graham, and some of the article’s calculations had been noticed with some (minimal) interest, by Professor Nash, during a very brief and casual discussion Princeton (in the corridor near the common room).
Professors Jaffe (Clay Harvard/MIT), Wiles, (Princeton), Farley (Harvard),Goins (Cal Tech, Stanford, Purdue), Massey (Princeton), Neerchal (Maryland), Scheinerman (Hopkins), Wierman (Hopkins), Hrabowski (Maryland), Conway (Princeton), Yoder (Maryland, Rockefeller, U.Mass), Greene (Columbia), Goldfeld (Columbia), Nash (Princeton) Goldberg (Queens), Huntley (Baruch), and others have been directly or indirectly helpful to me, even if not at all especially clear or interested in the topic of this article.
Helaman Ferguson had referred me back to John (Conway), after Professor Emeritus Paul Calter (Vermont) had invited me to attend a conference at Albany, after his (Paul’s) visit to see some of my work in New York.
Replies to my inquiries from Terence Jackson, from David Bressoud, from Erich Friedman, and from Neil Sloane have also helped me with this or with some of my other writing, as have those of Jerryl Payne(MIT, Cambridge, Lockheed-Martin).
This has been no less true of replies to me from Russ Rowlett, Eugene Pamfiloff, Ryan Harrison, F. Isadore Scott, Chediak, Griffin, Sanders, (all of Johns Hopkins) and from Professor Emeritus David G.S. Greene, (physics, Wisconsin, Johns Hopkins, Towson).
After defending the article with Professor Conway for several hours in the gardens at Princeton some years back, I have been planning to eventually polish the article, and to include in it more of some of the new calculations, which I had been able to prove to John, are indeed possible.
I will also want to explore a curious feature of one of the identities of Ramanujan, since some of my calculations have run into it in the article. I had first seen this identity in the biography of Ramanujan, by Professor Kanigel (MIT).
Finally, my point is that since as you say, “Many non-mathematicians have Erdös numbers as well”, mine (if I have one) may be either that of a mathematician, or of a non-mathematician, like Hank’s (although like Conway Graham, and Tao, unlike me, Hank’s Erdos number, and theirs, are 1).
While ever hopeful of eventually publishing some of my findings, I remain indebted to Professor David C. Tischler (Queens) for insisting that I must publish, during his discussion with me in his office, immediately after a most challenging mathematical thesis defense that he (Dave) had privately arranged for me.
Whether none, one, 7, 18, 32, or more published articles, let my comments here be a reminder of the many gifted mathematical thinkers, who from Ahmes and earlier, through Pythagoras, Euclid, Fibonacci, Pascal, Fermat, Newton, Euler, Gauss, Ramanujan, to Erdos, have and will continue to prove that the true greatness of the best mathematical minds is always most evident well beyond the shadows of their predecessors, of their contemporaries, and also well beyond the shadows of the most ingenious mathematical minds of the future.
Doc
R.A. Jones, MD. (SIAM)
Editor-in-Chief
The American Journal of Medical Mathematics
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Michael,
Thank you for posting my earlier comment.
Omitted in haste from it were Curtis Jones (Maryland, Westinghouse, Northrup Grumman, the first mathematician to check the derivation.
Also Professor Lemuel Moye, M.D., Ph.D (Johns Hopkins, Cornell, Texas), Professor Emeritus Julian Davis (Stevens Institute), Ernesto Vaterpool, Professor Jeffrey Shallit (Waterloo, MIT), Hany Farid (Dartmouth),Victor Shoup (Courant), and Mike Speciner (MIT) have all been helpful to me, even when quite busy, and or indifferent to some of topics I have been investigating.
If this will be easily done, kindly accept this ammendment to my reply.
Many thanks,
Doc (Ron)
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